Introduction

iGEM Heidelberg provides a comprehensive set of models that allows for both control and evaluation of continuous and discontinuous direction evolution. The interactive models facilitate regular use of the models in everyday lab work and are easier to understand as they provide an intuitive understanding by enabling the user to observe how the model behaves when parameters are changed. Predictions from the models helped to design the novel method Predcel to be both reliable and time efficient. To get accurate modelling results for the used setup, a selection of parameters was determined experimentally and included in the models. As models for different levels of abstraction were needed, a variety of approaches from ordinary differential equations, delayed differential equations over stochastic simulations to molecular dynamics was applied to obtain valuable information on the different aspects of directed evolution. Each term describing the change of an E. coli concentration contains its growth, \(g_{e}\). The growth rate of an E. coli population can be modelled by exponential growth or by logistic growth. Especially, when long durations per lagoon are modelled, the logistic growth model is more exact. [source]. In the exponential case the growth rate \(g_{e}\) is modelled as $$ g_{e} (t_{e}) = c_{e} \cdot \frac{log(2)}{t_{e} } $$ Note that the growth rate in the model increases over time, while in the modelled culture, the nutrient concentration decreases. That makes the logistic model more plausible, it models \(g_{e}\) as $$ g_{e} (t_{e}, \: c_{e}(t), \: c_{c}) = \frac{c_{c} - c_{e} (t)}{c_{c} } \cdot \frac{log(2)}{t_{e} } $$ In this case the learning rate decreases as the current concentration \(c_{e}\) approaches the maximum capacity for E. coli in the given setup \(c_{c}\). With this model \(c_{e} \leq c_{c}\) is true for any point in time. Change of concentration of uninfected E. coli, \(\frac{\partial c_{u} }{\partial t} \: [cfu/min]\) $$ \frac{\partial c_{u} }{\partial t}(t) = g_{u} (t_{u}, \: c_{u}(t), \: c_{c}) - k \cdot c_{u}(t) \cdot c_{p}(t) $$ In addition to the growth term, the concentration of uninfected E. coli is described by a term for infection that takes into account the concentration of uninfected E. coli and the concentration of free phage and reduces the conentration of uninfected E. coli. Change of concentration of uninfected E. coli, \(\frac{\partial c_{i} }{\partial t} \: [cfu/min]\) $$ \frac{\partial c_{i} }{\partial t}(t) = \begin{cases} g_{i} (t_{i}, \: c_{i}(t), \:c_{c}) + k \cdot c_{i}(t) \cdot c_{p}(t) - c_{i}(t - t_{P}), \quad \text{for} \: t > t_{P} \\ g_{i} (t_{i}, \: c_{i}(t), \: c_{c}) + k \cdot c_{i}(t) \cdot c_{p}(t), \quad \text{otherwise} \end{cases} $$ Until \(t > t_{P}\) the concentration of infected E. coli increases by growth and infection of previouly uninfected E. coli. When \(t > t_{P}\), a third term describing that infected E. coli turn into phage-producing E. coli is subtracted. Change of concentration of phage producing E. coli, \(\frac{\partial c_{p} }{\partial t} \: [cfu/min]\) $$ \frac{\partial c_{p} }{\partial t}(t) = \begin{cases} g_{p} (t_{p}, \: c_{p}(t), \: c_{c}) - c_{i}(t - t_{P}), \quad \text{for} \: t > t_{P} \\ g_{p} (t_{p}, \: c_{p}(t), \: c_{c}), \quad \text{otherwise} \end{cases} $$ The population of phage producing E. coli only increases by growth until \(t > t_{P}\). When infected E. coli drop their first phage they turn into producing E. coli as described by the second term. Change of concentration of M13 phage, \(\frac{\partial c_{P} }{\partial t} \: [cpu/min]\) $$ \frac{\partial c_{P} }{\partial t}(t) = c_{P}(t) \cdot \mu_{max} \cdot f - k \cdot c_{u}(t)\cdot c_{P}(t) $$ The phage concentration is only increased by phage that leave phage-producing E. coli, which happens at a rate of \(f \cdot \mu_{max}\) per time unit, with f being the fitness, a value between 0 and 1, equal to the share of the wildtype M13 phages fitness and \(\mu_{max}\) being the wildtype phages production rate. We assume that the only negative influence on the free phage titer is phage infecting E. coli, which depends on both the phage titer \(c_{P}\) and the titer of uninfected E. coli, \(c_{i}\). The fitness \(f\) is assumed to be constant during the time spent in one lagoon, it is assumed that all phages have the same fitness.

Modelling concentrations over multiple Lagoons

When transfer from one volume to the next is performed, new lagoon can be modelled with starting values calculated from the last lagoons end values. For each concentration from the previous lagoon \(c_{t}\), the concentration in the next lagoon \(c_{t+1}\) is calculated as $$ c_{t+1} = \frac{v_{t} }{v_{l} } \cdot c_{t} $$ with \(v_{l}\), the volume of a lagoon and \(v_{t}\), the volume that is transferred. If the transfered volume is spinned down before it is added to the new lagoon, the initial value for \(c_{P}\) is calculated this way. The initial concentration of uninfected E. coli is set to the initial cell density. Initial concentrations of infected and phage-producing E. coli are set to zero, because before the transfer, no phages are present in the new lagoon. If the transfer volume is not spinned down, the concentration of infected and phage-producing E. coli are calculated, using the above formula. The initial concentration of uninfected E. coli is the calculated the same way, but the initial cell density is added. In directed evolution the fitness should increase over time. A linear increase in fitness between to given values was implemented to show this. The problem with this approach is its basic assumption being that all phage-producing E. coli are infected by phages with the same fitness. To make the model more plausible, a distribution of fitness was introduced. For a set of discrete fitness values each fitness values share of the phage-producing E. coli population is calculated. That changes the equation for the change in the concentration of phage-producing E. coli to $$ \frac{\partial c_{P} (t)}{\partial t} = -k \cdot c_{u}(t) \cdot c_{P} (t) + \sum_{i = 0}^N f_{i} \cdot s_{i} \cdot \mu \cdot c_{p} (t) $$ The calculation is for \(N\) different fitness values \(f_{i}\) and their share of the total phage-producing E. coli population \(s_{i}\).

Numeric solutions

The problem described above is a system of four differential equations, of which two ( \(\frac{\partial c_{i} }{\partial t} \:, \: \frac{\partial c_{p} }{\partial t}\) ) are so called delayed differential equations. They contain a term that needs to be evaluated at a timepoint in the past \(t - t_{P}\). A custom script was used to solve the problem numerically, using the explicit Euler method.[Source!] The basic idea is that from a point in time with all values and all derivatives values given, the next point in time can be calculated by assuming a linear progress between the two points. $$ f(t_{n+1}) = f(t_{n}) + (t_{n+1} - t_{n}) \cdot f'(t_{n}) $$ This is performed for \(c_{u}(t)\), \(c_{i}(t)\), \(c_{p}(t)\) and \(c_{P}(t)\) rotatory, to always have the needed values from \(t_{n}\) ready for \(t_{n+1}\). To explore, how unprecise parameters and noise influence the outcome of the model, a mode was implemented, that adds gaussian noise to all parameters. It uses the function \(n\) that makes a value \(v\) noisy with a random parameter \(r\). $$ n(v) = \big(1 - 2r\big) \cdot \sigma_{G} \cdot \sigma_{v} \cdot v, \quad r \in (0, 1) $$ Here, \(\sigma_{G}\) is a factor that is the same for all \(v\), \(\sigma_{v}\) is specific for \(v\). This way, it is possible to have one parameter being noisier than another, while being able to tune the noise globally. [Results]